Micro-machined inertial sensors have become an integral part of many consumer products, such as hand-held mobile terminals, cameras, and game controllers. In addition, micro-machined inertial sensors are widely used in vibration monitoring in industry, automotive safety and stability control, and navigation. In general, the read-out mechanism of micro-sensors can be piezoelectric, piezoresistive or capacitive. However, the high thermal stability, and sensitivity of capacitive sensing makes it more attractive for wide range of applications. A typical capacitive sensor interface circuit is composed of a capacitance-to-voltage converter (C/V) followed by an analog-to-digital converter (A/D) and signal conditioning circuitry. Incorporating the sensor and C/V in a Σ-Δ based force feedback loop provides many benefits, such as reducing sensitivity to sensor process and temperature variation, improving system bandwidth, and increasing dynamic range. In addition, the Σ-Δ based loop provides implicit analog-to-digital conversion, eliminating the need for a stand-alone A/D.
Capacitive inertial MEMS sensors exhibit a second order low-pass transfer function. In some systems, the MEMS serves as the Σ-Δ loop filter, resulting in a second order electro-mechanical Σ-Δ loop. However, relying only on the MEMS as the filtering element results in a resolution penalty, due to increased quantization noise. The increased quantization noise results from the reduced effective Σ-Δ loop quantizer gain, caused by electronic noise. To avoid this resolution penalty an electronic filter may be introduced to the loop. The additional electronic filter can be of first or second order for accelerometer sensors (accelerometer sensors are devices used to measure linear acceleration). For certain gyroscope systems (gyroscope devices are used to measure the angular speed in degrees/sec), a second order filter is used to implement a resonator that can produce a notch in the noise transfer function away from DC, resulting in a fourth order modulator.
Ideally, capacitive inertial MEMS sensors would behave as a second-order lumped mass-damper-spring system, with a single resonant frequency. However, in reality the sensor is a distributed element that has additional parasitic resonant modes. These parasitic modes can lead to instability of the Σ-Δ modulator.
The following references (referred to hereinafter as Seeger, Petkov, and Ezekwe, respectively) address the problem of electro-mechanical Σ-Δ loop stability in the presence of parasitic modes:    J. I. Seeger. X. Jiang. M. Kraft, and B. E. Boser. “Sense Finger Dynamics in a Sigma Delta Force Feedback Gyroscope.” in Proc. Solid-State Sensor and Actuator Workshop Dig. Tech. Papers, June 2000, pp. 296-299.    V. P. Petkov, High-order Σ-Δ Interface for Micromachined Inertial Sensors. Dept. of Electrical Eng. and Comp. Science, UC Berkeley: Ph.D. Thesis, 2004.    C. D. Ezekwe, Readout Techniques for High-Q Micromachined Vibratory Rate Gyroscopes. Dept. of Electrical Eng. and Comp. Science, UC Berkeley: Ph.D. Thesis, 2007.
Stabilizing the loop in the presence of high-Q parasitic modes is a challenging problem. By some accounts, high-Q parasitic modes must be addressed with proper mechanical design, as using only electronic techniques was not successful [Petkov].
In Seeger, the stability of a second order electro-mechanical Σ-Δ loop is considered and it is suggested to maintain a certain relation between system sampling frequency and the parasitic mode frequency. However, Seeger is specific to a second order loop (a loop that does not incorporate electronic filter) with a low quality factor (Q) parasitic mode, and is not applicable to higher order loops or in the case of high-Q parasitic modes. In Petkov, on the other hand, the system was tested at atmospheric pressure, thereby ensuring that high frequency modes were sufficiently damped. In practice therefore, Petkov is applicable only to low-Q parasitic modes.
Ezekwe addresses high-Q parasitic modes. However, the proposed solution uses positive feedback techniques, resulting in nested feedback loops that are hard to design, optimize, and tune. More particularly, in Ezekwe, a positive feedback Σ-Δ loop is adopted. To avoid instability due to positive feedback, the DC gain is set below 1, by injecting a pseudo-random signal to the loop. Having the DC gain below 1 at DC makes the resulting system inadequate for accelerometers, and limits its use to gyroscopes, because, this condition reduces in-band noise attenuation. The loss of DC gain also results in accumulation of offset before the quantizer, which requires an additional regulation loop. The resulting system consists of nested loops that are hard to design, optimize, and tune.
An electronic fourth order Σ-Δ modulator (modulator has four integrators) with feed-forward summation is shown in FIG. 1. This electronic modulator may form the basis of electro-mechanical Σ-Δ capacitive interface circuits as described herein.